Problem: $z=5+7i$ Find the angle $\theta$ (in radians ) that $z$ makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express $\theta$ between $-\pi$ and $\pi$. $\theta=$
Answer: The strategy We can find the angle $\theta$ of any complex number $z$ by solving the following equation. $\tan\theta=\dfrac{\text{Im}(z)}{\text{Re}(z)}$ This equation usually has two solutions in the interval $[-\pi,\pi]$. We can find the appropriate solution by reasoning about the quadrant in which $z$ lies. Solving for $\theta$ $\begin{aligned}\tan\theta &= \dfrac{\text{Im}(z)}{\text{Re}(z)}\\\\ \tan\theta&=\dfrac{7}{5}\\\\ \theta&=\arctan\left(\dfrac{7}{5}\right)&\text{Take the arctangent of both sides}\\\\ \theta&\approx0.951\end{aligned}$ Using the identity $\tan(-\pi+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $-\pi+(0.951)=-2.191$ In order to determine which of these two solutions is the angle of $z$, let's take a look at its graphical representation. ${6}$ ${6}$ $Im$ $Re$ $z$ $\theta$ $Re(z)$ $Im(z)$ Since $z$ lies in Quadrant $\text{I}$, its angle must be in the interval $(0,\dfrac{\pi}{2})$. Therefore, $\theta=0.951$. Summary $\theta=0.951$